Re: Formulating sentences in a possibly consistent ZF

herbzet wrote:

Nam Nguyen wrote:
Aatu Koskensilta wrote:

[...]

Again, how sure are you about "entirely unproblematic"? For example,
if we can't know which one of GC and cGC is arithmetically true, then
we'd have to acknowledge that the notion about the naturals is
*subjective*, since you might take them to contain the truth of GC,
while I of cGC, right? And if arithmetic is subjective, then Godel's
assertions such as "G(Q) is true" is consequently subjective as well.
Right?

You might be interested to know that Goldbach's weak conjecture, that
every integer greater than 5 can be written as the sum of 3 primes,
is known to have, at most, a finite number of exceptions.

See http://En.wikipedia.org/wiki/Vinogradov%27s_theorem#A_consequence .

Of course, by "every integer greater than 5", it's meant to be
"every *odd* integer greater than 5". As such it seems make a huge
difference on the nature of difficulties between GC and wGC (the
waek GC mentioned above).

If we recall, an even number *could be purely defined* by a non-inductive
multiplicative way (as a product of primes one of which must be the smallest
prime). On the other hand, we can't define odd numbers in the same manner!

--
"To discover the proper approach to mathematical logic,
we must therefore examine the methods of the mathematician."
(Shoenfield, "Mathematical Logic")
.

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